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The exterior angles of a pentagon are x, 2x, x+30, 3x, and 50 degrees. Find x. The exterior angles of another pentagon are 70, 125, 2x, x+40, and 135 degrees. Find x.

Mathematics

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Answer

a

x=40∘x = 40^\circ

Step-by-step solution

8 steps

Part a· 4 steps

The diagram shows a pentagon. The sum of the exterior angles of any convex polygon is 360∘360^\circ.

  1. Identify the exterior angles from the diagram.

    The exterior angles are xx, 2x2x, x+30∘x+30^\circ, 3x3x, and 50∘50^\circ.

  2. Set up the equation by summing the exterior angles and equating them to 360∘360^\circ.

    x+2x+(x+30∘)+3x+50∘=360∘x + 2x + (x+30^\circ) + 3x + 50^\circ = 360^\circ

  3. Combine like terms.

    (x+2x+x+3x)+(30∘+50∘)=360∘(x+2x+x+3x) + (30^\circ+50^\circ) = 360^\circ 7x+80∘=360∘7x + 80^\circ = 360^\circ

  4. Solve for xx.

    7x=360∘−80∘7x = 360^\circ - 80^\circ 7x=280∘7x = 280^\circ x=280∘7x = \frac{280^\circ}{7} x=40∘\boxed{x = 40^\circ}

Part b· 4 steps

The diagram shows a pentagon. The sum of the exterior angles of any convex polygon is 360∘360^\circ.

  1. Identify the exterior angles from the diagram.

    The exterior angles are 70∘70^\circ, 125∘125^\circ, 2x2x, x+40∘x+40^\circ, and 135∘135^\circ.

  2. Set up the equation by summing the exterior angles and equating them to 360∘360^\circ.

    70∘+125∘+2x+(x+40∘)+135∘=360∘70^\circ + 125^\circ + 2x + (x+40^\circ) + 135^\circ = 360^\circ

  3. Combine like terms.

    (2x+x)+(70∘+125∘+40∘+135∘)=360∘(2x+x) + (70^\circ+125^\circ+40^\circ+135^\circ) = 360^\circ 3x+(195∘+40∘+135∘)=360∘3x + (195^\circ+40^\circ+135^\circ) = 360^\circ 3x+(235∘+135∘)=360∘3x + (235^\circ+135^\circ) = 360^\circ 3x+370∘=360∘3x + 370^\circ = 360^\circ

  4. Solve for xx.

    3x=360∘−370∘3x = 360^\circ - 370^\circ 3x=−10∘3x = -10^\circ
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